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The Champernowne constant[1] \(C_{10}\) is a real number whose decimal expansion is created by concatenating the decimal expansions of the positive integers:

C10 = 0.12345678910111213141516171819202122232425...

K. Mahler showed in 1961 that it is transcendental. It is also normal in base 10.

The continued fraction expansion (CFE) of the Champernowne constant turns out to be an unlikely source of large numbers, containing various spikes.[2] It begins 0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, and the coefficient in position 18 has 166 digits. The heuristic explanation for this is that the decimal expansion often runs very close to periodicity (e.g. ...142963142964142965142966...) and some of the convergents will be unusually close.

Even more surprisingly, in 2012 John Sikora detected patterns in these spikes and made several conjectures from observed regularities.[3] He calls CFE coefficients larger than any previous coefficients "high water marks" (HWMs):

  • HWM #1 = 0, position 0
  • HWM #2 = 8, position 1
  • HWM #3 = 9, position 2
  • HWM #4 = 149083, position 4
  • HWM #5 has 166 digits, position 18
  • HWM #6 has 2504 digits, position 40
  • HWM #7 has 33102 digits, position 162
  • HWM #8 has 4411100 digits, position 526
  • HWM #9 has 4911098 digits, position 1708
  • HWM #10 has 57111096 digits, position 4838
  • HWM #11 has 651111094 digits, position 13522
  • HWM #12 has 7311111092 digits, position 34062

The increasing sequences of 1's in the digit counts are examples of the patterns Sikora noticed.

We can also ask what happens to the CFE's of constants formed by concatenating other integer sequences. The "near-periodicity" argument predicts that the spikes will be less dramatic for faster-growing sequences. This matches Eric Weisstein's observation that the Copeland-Erdős constant, formed by concatenating the primes, has a better-behaved CFE — spikes are present, but smaller and less frequent.[4]

Sources Edit

  1. [1]
  2. [2]
  3. [3]
  4. Copeland-Erdős constant continued fraction

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